MR2668517 Nobu KISHIMOTO Nobu KISHIMOTO Counterexamples to Bilinear Estimates for the Korteweg-de Vries Equation in the Besov-Type Bourgain Space Funkcialaj Ekvacioj. Serio Internacia 53 2010 133--142 http://fe.math.kobe-u.ac.jp/FE/FullPapers/53-1/53_133.pdf http://www.ams.org/mathscinet-getitem?mr=MR2668517 Recently it was shown by Z. Guo and the author independently that the Cauchy problem for the nonperiodic Korteweg-de Vries equation is locally well-posed in the Sobolev space of the critical regularity. Their proofs were based on the iteration argument in the Besov endpoint of the Bourgain space with some additional modification in low frequency, and it was conjectured that the Bourgain space with the Besov-type modification only does not restore the bilinear estimate which is essential to the iteration. In the present article we give the answer to this conjecture by constructing an exact counterexample to the bilinear estimate. Korteweg-de Vries equation, Cauchy problem, Iteration method, Bilinear estimate, Bourgain space. 35Q53. 53-133 2010 Counterexamples to Bilinear Estimates for the Korteweg-de Vries Equation in the Besov-Type Bourgain Space Nobu KISHIMOTO Nobu KISHIMOTO
1 Bejenaru, I.; Tao, T. Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation J. Funct. Anal. 233 2006 228-259 MR2204680
1 Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolutions, Part I: Schrödinger equations Geom. Funct. Anal. 3 1993 107-156 MR1209299
2 Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlilnear evolutions, Part II: The KdV-equation Geom. Funct. Anal. 3 1993 209-262 MR1215780
1 Christ, M.; Colliander, J.; Tao, T. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations Amer. J. Math. 125 2003 1235-1293 MR2018661
1 Colliander, J.; Kenig, C. E.; Staffilani, G. Local well-posedness for dispersion-generalized Benjamin-Ono equations Differential Integral Equations 16 2003 1441-1472 MR2029909
1 Colliander, J.; Kenig, C. E.; Staffilani, G. Low regularity solutions for the Kadomtsev-Petviashvili I equation Geom. Funct. Anal. 13 2003 737-794 MR2006556
1 Guo, Z. Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(\mathbb{R})$ J. Math. Pures Appl. (9) 91 2009 583-597 MR2531556
5 Guo, Z., Local well-posedness and a priori bounds for the modified Benjamin-Ono equation without using a gauge transformation, preprint. (arXiv:0807.3764)
1 Hadac, M.; Herr, S.; Koch, H. Well-posedness and scattering for the KP-II equation in a critical space Ann. Inst. H. Poincaré Anal. Non Linéaire 26 2009 917-941 MR2526409
1 Ionescu, A. D.; Kenig, C. E. Low-regularity Schrödinger maps Differential Integral Equations 19 2006 1271-1300 MR2278007
1 Ionescu, A. D.; Kenig, C. E. Global well-posedness of the Benjamin-Ono equation in low-regularity spaces J. Amer. Math. Soc. 20 2007 753-798 MR2291918
1 Kenig, C. E.; Ponce, G.; Vega, L. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle Comm. Pure Appl. Math. 46 1993 527-620 MR1211741
17 Kenig, C. E.; Ponce, G.; Vega, L. The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices Duke Math. J. 71 1993 1-21 MR1230283
18 Kenig, C. E.; Ponce, G.; Vega, L. A bilinear estimate with applications to the KdV equation J. Amer. Math. Soc. 9 1996 573-603 MR1329387
1 Kenig, C. E.; Ponce, G.; Vega, L. On the ill-posedness of some canonical dispersive equations Duke Math. J. 106 2001 617-633 MR1813239
1 Kishimoto, N. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\overline{u}^2$ Commun. Pure Appl. Anal. 7 2008 1123-1143 MR2410871
1 Kishimoto, N. Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity Differential Integral Equations 22 2009 447-464 MR2501679
1 Muramatu, T.; Taoka, S. The initial value problem for the 1-D semilinear Schrödinger equation in Besov spaces J. Math. Soc. Japan 56 2004 853-888 MR2071676
1 Nakanishi, K.; Takaoka, H.; Tsutsumi, Y. Counterexamples to bilinear estimates related with the KdV equation and the nonlinear Schrödinger equation Methods Appl. Anal. 8 2001 569-578 MR1944182
1 Taoka, S. Well-posedness of the Cauchy problem for the semilinear Schrödinger equation with quadratic nonlinearity in Besov spaces Hokkaido Math. J. 34 2005 65-96 MR2130772
1 Tataru, D. Local and global results for wave maps, I Comm. Partial Differential Equations 23 1998 1781-1793 MR1641721